The minimum number of vertices in uniform hypergraphs with given ...
arXiv:1603.03557v1 [math.CO] 11 Mar 2016
The minimum number of vertices in uniform hypergraphs with given domination number ∗,† Csilla Bujt´as
‡,§
Bal´azs Patk´os
§
Zsolt Tuza
‡,§
M´at´e Vizer
§
March 14, 2016
Abstract The domination number γ(H) of a hypergraph H = (V (H), E(H)) is the minimum size of a subset D ⊂ V (H) of the vertices such that for every v ∈ V (H) \ D there exist a vertex d ∈ D and an edge H ∈ E(H) with v, d ∈ H. We address the problem of finding the minimum number n(k, γ) of vertices that a k-uniform hypergraph H can have if γ(H) ≥ γ and H does not contain isolated vertices. We prove that n(k, γ) = k + Θ(k 1−1/γ ) and also consider the s-wise dominating and the distance-l dominating version of the problem. In particular, we show that the minimum number ndc (k, γ, l) of vertices that a connected k-uniform hypergraph with distance-l domination number γ can have is roughly kγl 2 .
1
Introduction
In this paper we establish basic inequalities involving fundamental hypergraph parameters such as order, edge size, and domination number. Many problems in extremal combinatorics are of the following form: what is the smallest or largest size that a graph, hypergraph, set system can have, provided it satisfies a ∗
Research supported in part by the Hungarian Scientific Research Fund NKFIH/OTKA grant SNN 116095. The second author was also supported by the J´anos Bolyai Research Fellowship. † Authors’ email addresses: [email protected], [email protected], [email protected], [email protected] ‡ Department of Computer Science and Systems Technology, University of Pannonia, 8200 Veszpr´em, Egyetem u. 10, Hungary. § Alfr´ed R´enyi Institute of Mathematics, Hungarian Academy of Sciences, P.O.B. 127, Budapest H-1364, Hungary.
1
prescribed property? In most cases, size is measured by the number of edges, hyperedges, sets, respectively, contained in the object, and the number of vertices is usually included in the prescribed property. However, sometimes it can be interesting and even applicable to consider problems about the minimum or maximum number of vertices [18, 19, 20]. In the present paper we address the problem of finding the minimum number of vertices in a k-uniform hypergraph that has large domination number. The domination number γ(G) of a graph G, a widely studied notion (see [10], [11]), is the smallest size that a subset D ⊂ V (G) of the vertices can have if every vertex v ∈ V (G) \ D has a neighbor in D. We will be interested in the hypergraph version of this notion, which was investigated first in [1] and later studied in [2, 3, 4, 14, 16]. Let H = (V (H), S E(H)) be a hypergraph. The neighborhood 1 of a vertex v ∈ V (H) is the set Nv := {v} ∪ E∈E(H) : v∈E E, and the S neighborhood of a set S ⊂ V (H) is defined as N (S) := v∈S Nv . A set D ⊂ V (H) is called a dominating set of H if D ∩ Nv 6= ∅ for all v ∈ V (H). Equivalently we can say that D is a dominating set if and only if N (D) = V (H). The minimum size γ(H) of a dominating set in a hypergraph H is the domination number of H. As all isolated vertices always are contained in every dominating set, they can be eliminated in an obvious way, therefore we restrict our attention to hypergraphs without isolates. Let n(k, γ) be the minimum number of vertices that a k-uniform hypergraph with no isolated vertices must contain if its domination number is at least γ. Beyond the trivial case of n(k, 1) = k, the problem of determining n(k, γ) is natural and seems interesting enough to be addressed on its own right; neverhteless, Gerbner et al. (Problem 17 in [8]) arrived from a combinatorial search-theoretic framework at the particular problem of deciding whether n(k, 3) ≥ 2k + 3 holds or not. We answer this problem in the negative, determining the asymptotic behavior of n(k, γ) as a function of k for every fixed γ, up to the exact growth order of the second term. To state our result in full strength, we need to introduce two generalizations of domination. For an integer s > 0 we call D ⊂ V (H) an s-dominating set of H if |D ∩ Nv | ≥ s for all v ∈ V (H) \ D and we call D and s-tuple dominating set if |D ∩ Nv | ≥ s for all v ∈ V (H). Note that dominating sets are exactly the 1-dominating sets and 1-tuple dominating sets. As introduced in [7] and [9], respectively, the minimum size γ(H, s) of an s-dominating set in a hypergraph H is the s-domination number of H and the minimum size γ× (H, s) of an s-tuple dominating set in a hypergraph H is the s-tuple domination number 2 of H. By definition, we have γ(H, s) ≤ γ× (H, s). For every pair γ, s of integers with γ ≥ s, let n(k, γ, s) denote the minimum number of vertices that a k-uniform hypergraph H must have if γ(H, s) ≥ γ holds and there exist no isolated vertices in H and 1
In this paper we use the short term “neighborhood”, although this is called “closed neighborhood” in the main part of the literature. We note that the inclusion of {v} in the definition of Nv may be omitted if v is not an isolated vertex in H. 2 The standard notation for s-tuple domination in the graph theory literature is γ×s (G), but for the different variants of domination in this paper we try to use notations which are similar to each other in their form, this is why we put s in another position.
2
let n× (k, γ, s) denote the minimum number of vertices that a k-uniform hypergraph H must have if γ(H, s) ≥ γ holds and there exist no isolated vertices. From the above, we have n× (k, γ, s) ≤ n(k, γ, s). Our main theorem about s-domination is the following. Theorem 1.1. For every γ ≥ 2 and s ≥ 1 with γ > s we have k + k 1−1/(γ−s+1) ≤ n× (k, γ, s) ≤ n(k, γ, s) ≤ k + (4 + o(1))k 1−1/(γ−s+1) . Another generalization of domination is distance-l domination, which was introduced by Meir and Moon in [17]. This notion has been studied only for graphs so far. A good survey of the results until 1997 is [12]. For more recent upper and lower bounds on the distance-l domination number of graphs see [13] and [6]. In distance-l domination a vertex v dominates all vertices that are at distance at most l from v. As the definition of distance in graphs involves paths, and paths in hypergraphs can be defined in several ways, distance-l domination could be addressed with each of those definitions. But as we will remark in Section 4, only so-called ‘Berge paths’ offer new problems in our context. A Berge path of length l is a sequence v0 , H1 , v1 , H2 , v2 , . . . , Hl , vl with vi ∈ V (H) for i = 0, 1, ..., l and vi−1 , vi ∈ Hi ∈ E(H) for i = 1, 2, ..., l. The distance dH (u, v) of two vertices u, v ∈ V (H) is the length of a shortest Berge path from u to v. The ball centered at u and of radius l consists of those vertices of H which are at distance at most l from S u; it will be denoted by Bl (u). We call D ⊂ V (H) a distance-l dominating set of H if u∈D Bl (u) = V (H). Equivalently we can say that D ⊂ V (H) is a distance-l dominating set if and only if D ∩ Bl (v) 6= ∅ for all v ∈ V (H). Note that distance-1 dominating sets are the usual dominating sets. The minimum size of a distance-l dominating set in a hypergraph H is the distance-l domination number γd (H, l). Let further nd (k, γ, l) denote the minimum number of vertices that a k-uniform hypergraph H with no isolated vertices can contain if γd (H, l) ≥ γ holds. The next proposition shows that nd (k, γ, l) does not depend on l once l ≥ 2 is supposed. Proposition 1.2. For any k, l ≥ 2 and γ ≥ 1 we have nd (k, γ, l) = kγ, and the unique extremal hypergraph consists of γ pairwise disjoint edges. Proof. It is clear that the k-uniform hypergraph with just γ disjoint edges yields the upper bound nd (k, γ, l) ≤ kγ. We prove the lower bound by induction on γ. The case γ = 1 is trivial. So assume that γ ≥ 2, and let H = (V (H), E(H)) be a k-uniform hypergraph with γd (H, l) ≥ γ. Consider an arbitrary v ∈ V (H). Any vertex in N (Bl−1 (v)) is distance-l dominated by v, therefore the k-uniform hypergraph H0 induced by the edge set {H ∈ E(H) : H ∩ Bl−1 (v) = ∅} covers all vertices of H not distance-l dominated by v. The assumption γd (H, l) ≥ γ implies γd (H0 , l) ≥ γ − 1 and thus using that |Bl−1 (v)| ≥ k for l ≥ 2 and by induction we obtain |V (H)| = |Bl−1 (v)| + |V (H0 )| ≥ k + (γ − 1)k = γk. 3
Strict inequality holds whenever v has degree at least two. The problem becomes more interesting when disconnected hypergraphs get excluded. Hence, for k ≥ 2 and l, γ ≥ 1 let ndc (k, γ, l) denote the minimum number of vertices that a k-uniform connected hypergraph H must contain if it has γd (H, l) ≥ γ. To state our main result concerning ndc (k, γ, l) we need to define the following function: ( l kγ + max{k, γ} if l is even, f (k, γ, l) := 2l+1 kγ if l is odd. 2 Theorem 1.3. (a) For any k, l ≥ 2 we have (2l + 1)k (2l + 1)(k + 1) ≤ ndc (k, 2, l) ≤ min , (l + 1)k . 2 2 (b) For any k ≥ 2, l ≥ 4 and γ ≥ 3 we have l−1 − 1 γ < ndc (k, γ, l) ≤ f (k, γ, l). k 2 (c) For any k ≥ 2 and γ ≥ 3 we have kγ ≤ ndc (k, γ, 2) ≤ kγ + max{k, γ}. (d) For any k ≥ 2 and γ ≥ 3 we have kγ ≤ ndc (k, γ, 3) ≤ 2kγ.
The remainder of the paper is organized as follows: we prove Theorem 1.1 in Section 2, and Theorem 1.3 in Section 3. Section 4 contains some final remarks, also including a general upper bound on γdc (H, l) as a function of l, the number of vertices, and the edge size.
2
Proof of Theorem 1.1
In this section we prove our bounds on n× (k, γ, s) and n(k, γ, s). First we verify the bound k + k 1−1/(γ−s+1) ≤ n× (k, γ, s). Observe that it is enough to prove the statement for s = 1, since for any hypergraph H we have γ× (H, s) − (s − 1) ≥ γ× (H, 1) as for any s-tuple
4
dominating set D of H and a s0 -subset D0 of D the set D \ D0 (s − s0 )-tuple dominates H. Consequently n× (k, γ, s) ≥ n× (k, γ − (s − 1), 1), which implies the statement. To see n(k, γ, 1) ≥ k + k 1−1/γ let H be a k-uniform hypergraph with γ(H) ≥ γ ≥ 2. Let G = (V (H), E) be the graph with (u, v) ∈ E if and only if no H ∈ E(H) contains both u and v. The γ ≥ 2 condition means that for any vertex v ∈ V (H) there exists a u such that no edge H ∈ E(H) contains both u and v, thus G does not contain any isolated vertices. Let us write n = |V (H)| = |V (G)| = k + x and let t be the number of edges in a largest matching M = (V (M ), E(M )) of G. Note that two distinct vertices u0 , v 0 outside V (M ) cannot be adjacent to two distinct endpoints u, v of an edge e ∈ E(M ) as the matching (M \ {e}) ∪ {(u, u0 ), (v, v 0 )} would contradict the maximality of M . Then either just one of u and v has neighbors outside M , or none of them have any, or they share their unique neighbor outside M . We denote by e(v) the (or an) endpoint of e whose ‘outside’ neighborhood in this sense contains the ‘outside’ neighborhood of the other endpoint, and let de(v) denote the size of Ne(v) \ V (M ). By the definition of γ = γ× (H, 1) = γ(H, 1) and G we have that for any set Γ of γ − 1 vertices in V (G) there is a vertex v ∈ V (G) which is connected by edges in E(G) to all the vertices of Γ. If Γ is a subset of V (G) \ V (M ), then the vertex which is adjacent to all vertices of Γ must be in V (M ), since M is maximal. By this we obtain X de(v) |V (G) \ V (M )| ≥ . γ−1 γ−1 e∈E(M )
Writing d := maxe∈E(M ) de(v) the above inequality yields tdγ−1 ≥ (k + x − 2t)γ−1 , and rearranging gives d≥
k + x − 2t
. 1 t γ−1 Let e ∈ E(M ) be an edge with de(v) = d, and let H be any hyperedge H ∈ E(H) containing e(v). Just as any hyperedge, H must avoid an endpoint of each edge in M , and H is disjoint from Ne(v) \ V (M ). Therefore, we obtain k + x = n ≥ d + t + k and thus x ≥ d + t. Plugging the previous inequality into this and rearranging yields: γ−2
1
t γ−1 (x − t + 2t γ−1 ) ≥ k + x. γ−2
Now using that x ≥ t γand t ≥ t γ−1 , we obtain that the left-hand side of the previous inequality is at most x γ−1 + x and therefore we have γ
x γ−1 + x ≥ k + x, 5
which proves the required lower bound. To prove the bound n(k, γ, s) ≤ k + (4 + o(1))k 1−1/(γ−s+1) we need a construction. This involves projective geometries or linear vector spaces over finite fields. We will use the n Gaussian or q-binomial coefficient k q that denotes the number of k-dimensional subspaces of a vector space of dimension n over Fq , i.e. Qk n−i+1 − 1) n i=1 (q := Q k i k q i=1 (q − 1) and we will omit q from the subscript when it is clear from the context. Let q be a prime power, t be any positive integer and U be a γ-dimensional vector space over Fq . Let E1 , E2 , . . . , Em be the 1-dimensional of U and U1 , U2 , . . . , Um the γ-dimensional γsubspaces subspaces of U , where m = γ1 q = γ−1 = q γ−1 + q γ−2 + · · · + 1. Let A1 , A2 , . . . , Am , B be q pairwise disjoint sets with B = {b1 , b2 , . . . , bm } and |Ai | = t for all 1 ≤ i ≤ m. Let us define Hq,γ,t = {H1 , H2 , . . . , Hm } by [ Aj . Hi := {bi } ∪ j:Ei 6≤Uj
We claim that γ(Hq,γ−s+1,t , s) ≥ γ. Suppose not and let D = DB ∪ DA be a minimal sdominating set of H = Hq,γ−s+1,t with DB = D ∩ B, DA = D \ DB and |D| < γ. As every vertex d ∈ DB is contained in exactly one hyperedge Hd of H, each such d can be replaced by a vertex d0 ∈ V (H) \ (D ∪ B) to obtain an s-dominating set D0 with D0 ⊆ V (H) \ B 0 0 00 and |D T | = |D| < γ. Let D = {d1 , d2 , . . . , dp } and D = {d1 , d2 , . . . , dγ−s }. Then for Z = j:∃v∈D00 ∩Aj Uj we obtain dim(Z) ≥ 1. If E is a 1-subspace of Z, then the corresponding vertex b ∈ V (H) is not dominated by any vertex d ∈ D00 and thus at most (s − 1)-dominated by D0 , which is a contradiction. Let us consider the other parameters of the above hypergraph: n = |V (Hq,γ−s+1,t )| = m(t + 1) and Hq,γ−s+1,t is kq,γ−s+1,t -uniform with kq,γ−s+1,t = 1 + q γ−s t, therefore if t = q, then we obtain n = q γ−s+1 + 2(q γ−s + q γ−s−1 + · · · + q) + 1 and kγ = kq,γ−s+1,q = 1 + q γ−s+1 , 1−1/(γ−s+1) thus we have n ≤ kγ−s+1 + 4kγ−s+1 . This finishes the proof of the upper bound if k is one larger than the (γ − s + 1)st power of a prime. Finally, let us consider the general case when k 0 = 1 + q γ−s+1 + e with e < q 0γ−s+1 − q γ−s+1 where q 0 is the smallest prime larger than q. It is well-known that q 0 = q + o(q) and e e, all thus e = o(q γ−s+1 ). Let C1 , C2 , . . . , Cq+1 be pairwise disjoint sets all of size d q−γ+s+2 being disjoint from V (Hq,γ−s+1,q ). We renumber the subspaces U1 , U2 , . . . , Um in such a way that U1 , U2 , . . . , Uq+1 correspond to the dual of a (q + 1)-arc in P G(γ − s, q), i.e. every 1subspace E of V is contained in at most γ − s − 1 subspaces among U1 , U2 , . . . , Uq+1 . (For a general introduction to finite geometries, see [15].) Therefore, for any 1 ≤ i ≤ m, the sets 6
Ii := S {j : Ei 6≤ Uj , 1 ≤ j ≤ q + 1} satisfy |Ii | ≥ q − γ + s + 2 and thus there exists a set Ti ⊂ j∈Ii Cj of size e. Let us define [
Hi0 := {bi } ∪
Aj ∪ Ti .
j:Ei 6≤Uj
By definition we have |Hi0 | = k 0 for all i = 1, 2, . . . , m. The s-domination number of the new hypergraph is the same as that of the old one, as for any v ∈ Ci and u ∈ Ai we have Nu ⊂ Nv . Moreover the number n0 of vertices in the new hypergraph is n+d
e e(q + 1) ≤ k + 4k 1−1/(γ−s+1) + e + Oγ (e/q) ≤ k 0 + (4 + o(1))k 01−1/(γ−s+1) , q−γ+s+1
as Oγ (e/q) = o(q γ−s ) holds by e = o(q γ−s+1 ).
3
Distance domination
In this section we prove Theorem 1.3, the lower and upper bounds on ndc (k, γ, l).
3.1
The j-radius of trees
We start with some definitions and an auxiliary statement that we will use in the proof. Definition. For positive integers a1 , a2 , ..., ah the spider graph, denoted by S(a1 , a2 , . . . , ah ), P is the tree on 1 + hi=1 ai vertices which is obtained from h paths of lengths a1 , a2 , . . . , ah , respectively, by identifying the first vertices of those paths to a single vertex v of degree h. Hence, S(a1 , a2 , . . . , ah ) \ {v} has h connected components, say C1 , C2 , . . . , Ch , where each Ci is a path Pai on ai vertices (for i = 1, 2, ..., h). In a connected graph G = (V (G), E(G)), the excentricity of a vertex v ∈ V (G) is defined as excG (v) := max{dG (u, v) : u ∈ V (G)} and let the radius of G is r(G) := min{excG (v) : v ∈ V (G)}. More generally, for any ∅ = 6 W ⊂ V (G) let us write excG (W ) := max{min{dG (u, w) : w ∈ W } : u ∈ V (G)} 7
and for an integer j ≥ 1 let the j-radius of G be rj (G) := min{excG (W ) : W ⊂ V (G), |W | ≤ j}. Certainly we have r(G) = r1 (G). Finally, let rj (n) := max{rj (T ) : |V (T )| = n, T is a tree}. The numerical bounds themselves in the next lemma concerning the radius of a tree are folklore; for later use, however, we need a more detailed assertion which describes some structural properties, too. Some bounds on the function rj (n) can be derived also from results of Meir and Moon [17], but the following is a little sharper. Lemma 3.1. Let n ≥ j be positive integers. Then we have n n ≤ rj (n) ≤ . j+1 j+1 Moreover, r1 (n) = n−1 and 2 e is the path Pn on n vertices. (i) if n is even, then the only tree with r1 (T ) = d n−1 2 (ii) If n is odd and r1 (T ) = d n−1 e holds, then T is a path Pn−1 with a pendant edge. 2 Furthermore, T contains two copies of Pn−1 if and only if T is either a path Pn or a fork Fn . Otherwise T contains just one copy of Pn−1 . Proof. Let us first prove the statements about r1 (n). Let T be an arbitrary tree on n vertices and let v be a middle vertex of a longest path P in T . If P contains l vertices, then any e from v. This implies all assertions of the lemma if n is vertex is at distance at most d l−1 2 even. If n is odd, this implies that T must contain a path on n − 1 vertices and thus T is a path Pn−1 and a pendant edge. Let us now prove the general lower bound. We claim that n−1 n n+j−1 n = rj S , ,..., j+1 j+1 j+1 j+1 holds, which proves the lower bound by the definition of rj (n). To see that the claim is true, n observe that any set U ⊂ V (S(b n−1 c, b j+1 c, . . . , b n+j−1 c)) of size j is disjoint from at least j+1 j+1 n+j−1 n−1 n one component C of S(b j+1 c, b j+1 c, . . . , b γ+1 c) \ {v}. n c, b j+1 c, . . . , b n+j−1 c) belonging to C has distance Thus if v ∈ / U , then the leaf of S(b n−1 j+1 j+1 at least n−1 n 1+ ≥ j+1 j+1 8
from any vertex of U . If v ∈ U holds, then U is disjoint from at least two components C1 , C2 of n n S(b n−1 c, b j+1 c, . . . , b n+j−1 c) \ {v}, and the leaf of S(b n−1 c, b j+1 c, . . . , b n+j−1 c) belonging to j+1 j+1 j+1 j+1 n the larger path has distance at least b j+1 c from v and thus from U . This completes the proof of the general lower bound. To see the general upper bound, let T be any tree on n vertices. We will use the following claim repeatedly. Claim 3.2. Let m < n be two positive integers. Then in any tree T on n vertices there exists a vertex v such that if C1 , C2 , . . . , Cs denote Ps those components of T \ {v} whose all vertices are at distance at most m from v, then i=1 |Ci | ≥ m holds. Proof of Claim 3.2. Let P be a longest path of T . If P contains at most m vertices, then any vertex can play the role of v. If P contains at least m + 1 vertices, then let v be the (m + 1)st vertex from one end of P . For t = 1, 2, . . . , j − 1 let mt = b n+t−1 c and let T1 = T . We apply Claim 3.2 to Tt and j+1 mt for t = 1, 2, . . . , j − 1 to obtain vt ; and then set t Tt+1 := Tj \ ∪ki=1 Ci,t ,
where the Ci,t (i = 1, 2, ..., kt ) are the components of Tt \ {vt } whose vertices are at distance Pt at most mt from vt . By the claim we also have ki=1 |Ci,t | ≥ mt . n In this way we obtain a tree Tj of at most d2 j+1 e vertices. Let vj be a vertex of Tj |V (T )|−1
j e from all vertices of Tj . Such a vertex exists by the result on within distance d 2 n r1 (n). Clearly, U = {v1 , v2 , . . . , vj } is a set of vertices with excT (U ) ≤ d j+1 e, which proves n rj (T ) ≤ d j+1 e.
3.2
Putting things together: the proof of Theorem 1.3
Let us first prove the upper bounds of Theorem 1.3. To do so we introduce two types of hypergraphs with distance-l domination number γ. The second construction will prove the upper bounds of (b), (c), and (d). If γ = 2, then the construction giving the smaller number of vertices depends on the values of k and l. This is why we have the minimum of two expressions in the upper bound of (a). Construction 1: For i = 1, . . . , 2l(γ − 1) + 1 let Ui be pairwise disjoint sets, and let vi and w be distinct S vertices which are not elements of 2l(γ−1)+1 Ui . During Construction 1 all the indices will i=1 be taken modulo 2l(γ − 1) + 1, e.g. we then have 2l(γ − 1) + 2 = 1.
9
If k is odd, let |Ui | = following way. Let
k−1 2
for all i. We define a hypergraph H = (V (H), E(H)) in the 2l(γ−1)+1
V (H) :=
[
(Ui ∪ {vi }),
i=1
and let the hyperedges of H be Hi := Ui ∪ Ui+1 ∪ {vi } for i = 1, . . . , 2l(γ − 1) + 1. Then the size of V (H) is (2l(γ − 1) + 1)(k + 1) . 2 If k is even, let |U2i | = k2 for i = 1, . . . , l(γ − 1) and |U2i+1 | = We define H with the vertex set
k 2
− 1 for i = 0, . . . , l(γ − 1).
2l(γ−1)+1
V (H) := {w} ∪
[
(Ui ∪ {vi }),
i=1
and with the edge set E(H) := {Hi | 1 ≤ i ≤ 2l(γ − 1) + 1}, where Hi := Ui ∪ Ui+1 ∪ {vi } if i = 1, . . . , 2l(γ − 1), and Hi := Ui ∪ Ui+1 ∪ {vi , w} if i = 2l(γ − 1) + 1. Then, (2l(γ − 1) + 1)(k + 1) 1 (2l(γ − 1) + 1)(k + 1) |V (H)| = + = . 2 2 2 To see that γd (H, l) ≥ γ holds in both cases, observe the following facts: • vertex vi distance-l dominates a vertex vj exactly for j ∈ {i − l + 1, ..., i + l − 1}, • vertex w distance-l dominates a vj exactly for j ∈ {2l(γ − 1) − l + 2, ..., 2l(γ − 1) + l}, • a vertex u ∈ Ui distance-l dominates a vj exactly for j ∈ {i − l, ..., i + l − 1}. 10
So, every vertex in V (H) distance-l dominates at most 2l vertices vi . This yields γd (H, l) ≥ γ. Construction 2: This construction relies on the spider graph S = S(a1 , a2 , . . . , aγ ) with all of the ai being equal to bl/2c. Let v be the only vertex of S with degree γ. Let u1 , u2 , . . . , uγ be the neighbors of v in S, and let u01 , u02 , . . . , u0γ be the vertices of S that are at distamce bl/2c from v. Let W be a set of size max{k, γ}. Take a partition (W1 , W2 , . . . , Wγ ) of W such that |Wi | = b |W |+i−1 c. Finally, for every u ∈ V (S) \ {v}, let Uu := Uu,1 ∪˙ Uu,2 be a set of size k γ such that • u ∈ Uu,1 holds for all u ∈ V (S) \ {v}, • Uu ∩ Uu0 = ∅ holds for all u 6= u0 ∈ V (S) \ {v}, • Uu ∩ W = ∅ holds for all u ∈ V (S) \ {v}, • |Uu,1 | = |Wi | for all those u ∈ V (S) \ {v} which lie in the same component of S \ {v} as ui . With the help of the previously defined sets we construct a k-uniform hypergraph H in the following way, depending on the parity of l: Case I: l is even Let the vertex set of H be V (H) = W ∪ |V (H)| =
S
u∈V (S)\{v}
Uu . Thus we have
klγ + max{k, γ}. 2
The edge set E(H) contains the following four types of hyperedges: 1. all k-subsets of W , i.e. Wk ⊂ E(H), 2. for all u ∈ V (S) \ {v}, we have Uu ∈ E(H), 3. for all i = 1, 2, . . . , γ let Wi ∪ Uui ,2 ∈ E(H), 4. for every edge (u, u0 ) = e ∈ E(S) with u, u0 6= v if dS (u, v) < dS (u0 , v) holds, then let Uu,1 ∪ Uu0 ,2 ∈ E(H). Clearly, H is connected due to Wk ⊂ E(H). We claim that γd (H, l) ≥ γ holds. Indeed, if D ⊂ V (H) has size at most γ − 1, then there exists an i ≤ γ such that [ D ∩ (Wi ∪ Uu ) = ∅ u∈Ci
11
Figure 1: Construction 2 in case of even l
holds where Ci is the component of S \ {v} containing ui . Then u0i is at distance at least 1 + 2 2l = l + 1 from any vertex of D and thus u0i is not distance l-dominated by D. Case II: l is odd In addition to the sets defined above, let Z1 , Z2 , . . . , Zγ be pairwise disjont sets of size k − |Wi |, each of which is disjoint from all previously defined sets. Let the vertex set of H be γ [ [ V (H) = W ∪ Uu ∪ Zi . u∈V (S)\{v}
i=1
Thus we have
l |V (H)| ≤ d ekγ. 2 As for the edge set of H, there is a fifth type of hyperedge: 5. for all 1 ≤ i ≤ γ let Uu0i ,1 ∪ Zi ∈ E(H). The fact that γd (H, l) ≥ γ follows similarly as in the previous case, because for any (γ −1)-set D ⊂ V (H) there exists an i such that any vertex z ∈ Zi is at distance at least l + 1 from D. Let us now turn our attention to the lower bounds. We prove first that of (a). Consider a connected k-uniform hypergraph H with γd (H, l) ≥ 2. Let M be a maximal matching in 12
H obtained in the following way. Let M1 := {H1 }, I1 := {H ∈ E(H) \ {H1 } : H ∩ H1 6= ∅} and R0 := E(H) \ (M1 ∪ I1 ). Then for s ≥ 2 we define a sequence Ms , Is , Rs of partitions of E(H) such that: 1. Ms is a matching, 2. every hyperedge in Is meets at least one hyperedge in Ms , and 3. all hyperedges in Rs are disjoint from all hyperedges in Ms . If Ms , Is , Rs are defined with Rs 6= ∅, then let Hs+1 ∈ Rs be a hyperedge such that Hs+1 ∩ Is 6= ∅ for some Is ∈ Is . The existence of such Hs+1 follows from the assumption that H is connected. Set Ms+1 := Ms ∪ {Hs+1 }, Is+1 := Is ∪ {R ∈ Rs \ {Hs+1 } : R ∩ Hs+1 6= ∅} and Rs+1 := E(H) \ (Ms+1 ∪ Is+1 ). For the smallest positive t with Rt = ∅, we let M := Mt . Thus the size of M is t. Now let us consider the auxiliary graph GM with vertex set M and e = {Hi , Hj } ∈ E(GM ) if and only if there exists H ∈ H with H ∩ HS i 6= ∅ and H ∩ Hj 6= ∅. By the definition of M, the graph GM is connected. For a vertex v ∈ H∈M H let Hv denote the only element of M containing v. Suppose that for a pair H, H 0 ∈ M we have dGM (H, H 0 ) = r. Then for any pair of vertices u ∈ H, v ∈ H 0 we have dH (u, v) ≤ 1 + 2r. To see this, consider the sequence H, He1 , Hi1 , He2 , Hi2 , . . . , Her , H 0 , where es is the sth edge in a shortest path from H to H 0 and His is the sth vertex (i.e. a hyperedge in H) in the same path. By the maximality of M, for every vertex w of H there exists an edge Hw containing w and an edge H ∈ M with Hw ∩ H 6= ∅, therefore by the observation above we have dH (u, w) ≤ 2 + 2rGM (u) for every u ∈
S
H∈M
H and w ∈ V (H).
If t ≥ l + 1 holds, then |V (H)| ≥ kt ≥ k(l + 1), proving the desired lower bound. Now suppose that t ≤ l − 2 or t = l − 1 with t being odd. As we have noted, GM is connected and thus by Lemma 3.1 we obtain t−1 . r(GM ) ≤ 2 e holds and so, by Therefore, there exists an H ∗ ∈ M = V (GM ) such that rGM (H ∗ ) ≤ d t−1 2 ∗ the above, for a vertex v ∈ H we have t−1 0 dH (v, v ) ≤ 2 + 2 2 13
for any vertex v 0 ∈ V (H). So in this case a vertex v ∈ H ∗ distance-l dominates H, contradicting γd (H, l) ≥ 2. If t = l − 1 and t is even, then let T be a spanning tree of GM . By Lemma 3.1 we obtain that T is a path on t vertices. So we may assume that E(GM ) ⊃ {(Hi , Hi+1 ) : i = 1, . . . , t − 1}. Let e = (Ht/2 , Ht/2+1 ) and consider a vertex v ∈ Ht/2 ∩He . As for vertices v 0 with Hv0 ∩Hi 6= ∅ for some i > t/2, a shortest path in H between v and v 0 need not contain Ht/2 . Thus we obtain that v distance-l dominates H, contradicting γd (H, l) ≥ 2. Finally, it remains S to prove the lower bound of (a) in case of t = l and thus it is enough to prove that |V (H) \ H∈M H| ≥ k/2 holds. We may and will assume that the radius of GM is d l−1 e. Let T be a spanning tree of GM . By Lemma 3.1 we know that T is a path if l is even, 2 and T contains a path on l − 1 vertices if l is odd. We claim that even if l is odd, T must be a path on t vertices. Indeed, otherwise any vertex v ∈ He distance-l dominates H where e is the middle edge of a path on l − 1 vertices that is contained in T . This would contradict γd (H, l) ≥ 2. By this we may assume that E(GM ) ⊃ {(Hi , Hi+1 ) : i = 1, . . . , l − 1}. Claim 3.3. We have the following: (i) For any pair of edges e, e0 in T we have He ∩ He0 = ∅. S (ii) There exist w, w0 ∈ V (H) \ H∈M H and Hw , Hw0 ∈ E(H) with w ∈ Hw and w0 ∈ Hw0 , such that Hw meets only H1 and Hw0 meets only Hl , moreover Hw and Hw0 are disjoint from all the other H ∈ M and also from He for all e ∈ E(T ). Proof of Claim. We have two cases depending on the parity of l. Case I: l is even. Now we prove (i) in this case. Suppose that Hei ∩ Hej 6= ∅ with ei = (Hi , Hi+1 ), ej = (Hj , Hj+1 ). If i < j ≤ l/2, then a vertex v ∈ Hel/2 ∩ Hl/2+1 distance-l dominates H, contradicting γdist (H, l) ≥ 2.. Similarly, if i < j and j ≥ l/2, then a vertex v ∈ Hel/2 ∩ Hl/2 distance-l dominates H, contradicting γdist (H, l) ≥ 2. Also, if i < l/2 < j, then if l/2 − i ≤ j −l/2, then a vertex v from Hl/2−1 ∩Hel/2−1 distance-l dominates H, while if l/2−i ≥ j −l/2, then a vertex v from Hl/2+2 ∩ Hel/2+1 distance-l dominates H, contradicting γdist (H, l) ≥ 2. We are done with (i) in Case I. S To see (ii) suppose that, for every w ∈ V (H) \ H∈M H and Hw containing w, the hyperedge Hw meets He for some e ∈ E(T ) or Hw meets some Hz with z ≥ 2. Then a vertex in Hen/2 ∩ Hn/2+1 distance-l dominates H, contradicting γdist (H, l) ≥ 2. The existence of w0 and Hw0 can be shown analogously. This proves (ii) in Case I. 14
Case II: l is odd. The proof of this case is S very similar to the previous one. Let us just show (ii). Suppose that, for every w ∈ V (H) \ H∈M H and Hw containing w, the hyperedge Hw meets He for some e ∈ E(T ) or Hw meets some Hz with z ≥ 2. Then a vertex in Hedn/2e ∩ Hdn/2e distance-l dominates H, contradicting γdist (H, l) ≥ 2. S S Note that Hw ∩ Hw0 ⊂ V (H) \ H∈M H and also Hw ∪ Hw0 ∪ e∈E(T ) He ⊂ V (H), and thus writing I = |Hw ∩ Hw0 | we obtain |V (H)| ≥ max{lk + I, (l + 1)k − I} ≥ lk + k/2. This finishes the proof of the lower bound of (a). Next we prove the lower bound of (b). We will need the following lemma. Lemma 3.4. For any γ, l ≥ 2, let t∗ denote smallest t with rγ−1 (t) ≥
l−1 . 2
Then we have
ndc (k, γ, l) ≥ t∗ k. Proof. Let H be a connected k-uniform hypergraph with γd (H, l) ≥ γ. Let M be a maximal matching in H obtained as in the proof of the lower bound of part (a), and let us consider the auxiliary graph GM . For a vertex v ∈ ∪H∈M H let Hv denote the only element of M . containing v. Let the size of M be t. We assume first that t < t∗ , what means rγ−1 (t) < l−1 2 Suppose that for a pair H, H 0 ∈ M we have dGM (H, H 0 ) = r. Then for any pair of vertices u ∈ H, v ∈ H 0 we have dH (u, v) ≤ 1 + 2r. To see this, consider the sequence H, He1 , Hi1 , He2 , Hi2 , . . . , Her , H 0 , where es is the sth edge in a shortest path from H to H 0 and His is the sth vertex (i.e. a hyperedge in H) in the same path. Let U ⊂ M be a subset of size γ − 1 with rGM (U) = rγ−1 (GM ) ≤ rγ−1 (t), and let L ⊂ V (H) be a set containing one vertex from each U ∈ U. By the maximality of M, for every vertex w of H there exists an edge Hw containing w and an edge H ∈ M with Hw ∩ H 6= ∅. Therefore by the observation above and by the definition of U, there exists a U ∈ U and a vertex u ∈ U for which we have dH (u, w) ≤ 2 + 2rGM (U ) ≤ 2 + 2rγ−1 (t) < 2 + 2
l−1 = l + 1. 2
This means that if t < t∗ holds, then the (γ − 1)-subset L distance-l dominates H. Therefore M consists of at least t∗ hyperedges and thus |V (H)| ≥ t∗ k holds. The lower bound of (b) follows by applying Lemma 3.1 with j = γ − 1 together with Lemma 3.4, noting that d γt e ≥ l−1 implies γt > l−1 − 1. 2 2 Finally, we prove the lower bound of (c) and (d). This will follow from the claim that any maximal matching in the edge set E(H) of a connected hypergraph H with γd (H, 2) ≥ γ has size at least γ. To see this suppose that M = {H1 , H2 , . . . , Hm } is a maximal matching 15
in E(H) and for any i = 1, 2, . . . , m let vi be a vertex of Hi . As any vertex v ∈ V (H) is contained in a hyperedge Hv which, by maximality of M, intersects some Hi ∈ M, the set D = {vi : i = 1, 2, . . . , m} distance-2 dominates H. Therefore m ≥ γ must hold as claimed.
4
Final remarks and open problems
We addressed the problem of finding the minimum number of vertices that a connected kuniform hypergraph with high domination number must contain, and we considered two main variants of the problem. For the original notion of domination and for s-wise domination we found general lower and upper bounds on n(k, γ, s) in which even the order of magnitude of the second term matches. The natural open problem occurs: it can be of interest to find the constant coefficient of this second term. Theorem 1.3, our main result concerning distance domination determines the asymptotics of ndc (k, γ, l) if k and γ are fixed and l tends to infinity, or if all three parameters tend to infinity. Closing the gap of roughly 2kγ between the upper and lower bounds remains an interesting open problem. We had a good reason to choose the notion of Berge paths in the definition of distance-l domination. The most common other definitions of a path in hypergraphs are linear paths, where two consecutive hyperedges of the path must share exactly one vertex (an even more restrictive notion is a loose path) and tight paths where the vertices v1 , v2 , . . . , vk+l−1 of the path should be chosen in such a way that the ith hyperedge of the path is {vi , vi+1 , . . . , vi+k−1 } for all i = 1, 2, . . . , l. This implies that consecutive hyperedges of a tight path share k − 1 vertices. Note that in the construction showing the upper bound of Theorem 1.1 no pair of hyperedges has intersection size 1 or k − 1, therefore the construction does not contain linear or tight paths of length larger than 1 and thus distance domination would not differ from ordinary domination, had we used these notions of hypergraph paths to define distance. There are various results on different domination numbers of a hypergraph in the literature: on the s-domination number in [2], on the inverse domination number in [16], on the total domination number in [5], and on the connection of the domination number with the transversal number in [3], [4]. Let us finish with the following theorem that can be obtained simply by rearranging the lower bound of Theorem 1.3. In the style of Meir and Moon [17], it uses only the size of the vertex set, the prescribed distance bound l, and the uniformity of H. Theorem 4.1. If H is a connected k-uniform hypegraph with |V (H)| = n, then ( n if l = 2, 3 or 4, γdc (H, l) ≤ nk 2 · if l > 4. k l−3 It remains an open problem to make these upper bounds tight. 16
References [1] B. D. Acharya, Domination in hypergraphs, AKCE International Journal of Graphs and Combinatorics, 4 (2007), pp. 117–126. [2] B. D. Acharya, Domination in hypergraphs II, New directions, Proceedings of ICDM, Mysore, India, 2008, Ramanujan Mathematical Society Lecture Notes Series, 13 (2010), pp. 1–18. ´ s, and Zs. Tuza, Equality of domination [3] S. Arumugam, B. K. Jose, Cs. Bujta and transversal numbers in hypergraphs, Discrete Applied Mathematics, 161 (2013), pp. 1859–1867. ´ s, M. A. Henning, and Zs. Tuza, Transversals and domination in uni[4] Cs. Bujta form hypergraphs, European Journal of Combinatorics, 33 (2012), pp. 62–71. ´ s, M. A. Henning, Zs. Tuza, and A. Yeo, Total transversals and total [5] Cs. Bujta domination in uniform hypergraphs, The Electronic Journal of Combinatorics, 21(2014), #P2.24. [6] R. Davila, C. Fast, M. A. Henning, and F. Kenter, Lower bounds on the distance domination number of a graph, arXiv:1507.08745 [7] J. F. Fink and M. S. Jacobson, On n-domination, n-dependence and forbidden subgraphs, In: Graph Theory with Applications to Algorithms and Computer Science, Wiley, New York (1985), pp. 301–311. ´ lvo ¨ lgyi, B. Patko ´ s, M. Vizer, and G. [8] D. Gerbner, B. Keszegh, D. Pa Wiener, Finding a majority ball with majority answers, arXiv:1509.08276 [9] F. Harary and T. W. Haynes, Nordhaus-Gaddum inequalities for domination in graphs, Discrete Mathematics, 155 (1996), pp. 99-105. [10] T. W. Haynes, S. T. Hedetniemi, and P. J. Slater (eds), Fundamentals of Domination in Graphs, Marcel Dekker, Inc. New York, 1998. [11] T. W. Haynes, S. T. Hedetniemi, and P. J. Slater (eds), Domination in Graphs: Advanced Topics, Marcel Dekker, Inc. New York, 1998. [12] M. A. Henning, Distance domination in graphs, Domination in Graphs: Advanced Topics, T.W. Haynes, S.T. Hedetniemi, and P.J. Slater (eds), Marcel Dekker, Inc. New York, (1998), pp. 335–365.
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[13] M. A. Henning, N. Lichiardopol, Distance domination in graphs with given minimum and maximum degree, manuscript. ¨ wenstein, Hypergraphs with large domination number and [14] M. A. Henning, C. Lo with edge sizes at least three, Discrete Applied Mathematics, 160 (2012), pp. 1757–1765. [15] J. W. P. Hirschfeld, Projective geometries over finite fields, Clarendon Press, Oxford, 1979, 2nd edition, 1998. [16] B. K. Jose, Zs. Tuza, Hypergraph domination and strong independence, Applicable Analysis and Discrete Mathematics, 3 (2009), pp. 347–358. [17] A. Meir and J. W. Moon, Relations between packing and covering number of a tree, Pacific Journal of Mathematics, 61 (1975), pp. 225–233. ´ s, On the number of maximal intersecting k-uniform families and [18] Z.L. Nagy, B. Patko further applications of Tuza’s set pair method, The Electronic Journal of Combinatorics, 22 (2015), #P1.83. [19] Zs. Tuza, Critical hypergraphs and intersecting set-pair systems, Journal of Combinatorial Theory, Series B, 39 (1985), pp. 134–145. [20] Zs. Tuza, Inequalities for two set systems with prescribed intersections, Graphs and Combinatorics, 3 (1987), pp. 75–80.
18
The minimum number of vertices in uniform hypergraphs with given domination number ∗,† Csilla Bujt´as
‡,§
Bal´azs Patk´os
§
Zsolt Tuza
‡,§
M´at´e Vizer
§
March 14, 2016
Abstract The domination number γ(H) of a hypergraph H = (V (H), E(H)) is the minimum size of a subset D ⊂ V (H) of the vertices such that for every v ∈ V (H) \ D there exist a vertex d ∈ D and an edge H ∈ E(H) with v, d ∈ H. We address the problem of finding the minimum number n(k, γ) of vertices that a k-uniform hypergraph H can have if γ(H) ≥ γ and H does not contain isolated vertices. We prove that n(k, γ) = k + Θ(k 1−1/γ ) and also consider the s-wise dominating and the distance-l dominating version of the problem. In particular, we show that the minimum number ndc (k, γ, l) of vertices that a connected k-uniform hypergraph with distance-l domination number γ can have is roughly kγl 2 .
1
Introduction
In this paper we establish basic inequalities involving fundamental hypergraph parameters such as order, edge size, and domination number. Many problems in extremal combinatorics are of the following form: what is the smallest or largest size that a graph, hypergraph, set system can have, provided it satisfies a ∗
Research supported in part by the Hungarian Scientific Research Fund NKFIH/OTKA grant SNN 116095. The second author was also supported by the J´anos Bolyai Research Fellowship. † Authors’ email addresses: [email protected], [email protected], [email protected], [email protected] ‡ Department of Computer Science and Systems Technology, University of Pannonia, 8200 Veszpr´em, Egyetem u. 10, Hungary. § Alfr´ed R´enyi Institute of Mathematics, Hungarian Academy of Sciences, P.O.B. 127, Budapest H-1364, Hungary.
1
prescribed property? In most cases, size is measured by the number of edges, hyperedges, sets, respectively, contained in the object, and the number of vertices is usually included in the prescribed property. However, sometimes it can be interesting and even applicable to consider problems about the minimum or maximum number of vertices [18, 19, 20]. In the present paper we address the problem of finding the minimum number of vertices in a k-uniform hypergraph that has large domination number. The domination number γ(G) of a graph G, a widely studied notion (see [10], [11]), is the smallest size that a subset D ⊂ V (G) of the vertices can have if every vertex v ∈ V (G) \ D has a neighbor in D. We will be interested in the hypergraph version of this notion, which was investigated first in [1] and later studied in [2, 3, 4, 14, 16]. Let H = (V (H), S E(H)) be a hypergraph. The neighborhood 1 of a vertex v ∈ V (H) is the set Nv := {v} ∪ E∈E(H) : v∈E E, and the S neighborhood of a set S ⊂ V (H) is defined as N (S) := v∈S Nv . A set D ⊂ V (H) is called a dominating set of H if D ∩ Nv 6= ∅ for all v ∈ V (H). Equivalently we can say that D is a dominating set if and only if N (D) = V (H). The minimum size γ(H) of a dominating set in a hypergraph H is the domination number of H. As all isolated vertices always are contained in every dominating set, they can be eliminated in an obvious way, therefore we restrict our attention to hypergraphs without isolates. Let n(k, γ) be the minimum number of vertices that a k-uniform hypergraph with no isolated vertices must contain if its domination number is at least γ. Beyond the trivial case of n(k, 1) = k, the problem of determining n(k, γ) is natural and seems interesting enough to be addressed on its own right; neverhteless, Gerbner et al. (Problem 17 in [8]) arrived from a combinatorial search-theoretic framework at the particular problem of deciding whether n(k, 3) ≥ 2k + 3 holds or not. We answer this problem in the negative, determining the asymptotic behavior of n(k, γ) as a function of k for every fixed γ, up to the exact growth order of the second term. To state our result in full strength, we need to introduce two generalizations of domination. For an integer s > 0 we call D ⊂ V (H) an s-dominating set of H if |D ∩ Nv | ≥ s for all v ∈ V (H) \ D and we call D and s-tuple dominating set if |D ∩ Nv | ≥ s for all v ∈ V (H). Note that dominating sets are exactly the 1-dominating sets and 1-tuple dominating sets. As introduced in [7] and [9], respectively, the minimum size γ(H, s) of an s-dominating set in a hypergraph H is the s-domination number of H and the minimum size γ× (H, s) of an s-tuple dominating set in a hypergraph H is the s-tuple domination number 2 of H. By definition, we have γ(H, s) ≤ γ× (H, s). For every pair γ, s of integers with γ ≥ s, let n(k, γ, s) denote the minimum number of vertices that a k-uniform hypergraph H must have if γ(H, s) ≥ γ holds and there exist no isolated vertices in H and 1
In this paper we use the short term “neighborhood”, although this is called “closed neighborhood” in the main part of the literature. We note that the inclusion of {v} in the definition of Nv may be omitted if v is not an isolated vertex in H. 2 The standard notation for s-tuple domination in the graph theory literature is γ×s (G), but for the different variants of domination in this paper we try to use notations which are similar to each other in their form, this is why we put s in another position.
2
let n× (k, γ, s) denote the minimum number of vertices that a k-uniform hypergraph H must have if γ(H, s) ≥ γ holds and there exist no isolated vertices. From the above, we have n× (k, γ, s) ≤ n(k, γ, s). Our main theorem about s-domination is the following. Theorem 1.1. For every γ ≥ 2 and s ≥ 1 with γ > s we have k + k 1−1/(γ−s+1) ≤ n× (k, γ, s) ≤ n(k, γ, s) ≤ k + (4 + o(1))k 1−1/(γ−s+1) . Another generalization of domination is distance-l domination, which was introduced by Meir and Moon in [17]. This notion has been studied only for graphs so far. A good survey of the results until 1997 is [12]. For more recent upper and lower bounds on the distance-l domination number of graphs see [13] and [6]. In distance-l domination a vertex v dominates all vertices that are at distance at most l from v. As the definition of distance in graphs involves paths, and paths in hypergraphs can be defined in several ways, distance-l domination could be addressed with each of those definitions. But as we will remark in Section 4, only so-called ‘Berge paths’ offer new problems in our context. A Berge path of length l is a sequence v0 , H1 , v1 , H2 , v2 , . . . , Hl , vl with vi ∈ V (H) for i = 0, 1, ..., l and vi−1 , vi ∈ Hi ∈ E(H) for i = 1, 2, ..., l. The distance dH (u, v) of two vertices u, v ∈ V (H) is the length of a shortest Berge path from u to v. The ball centered at u and of radius l consists of those vertices of H which are at distance at most l from S u; it will be denoted by Bl (u). We call D ⊂ V (H) a distance-l dominating set of H if u∈D Bl (u) = V (H). Equivalently we can say that D ⊂ V (H) is a distance-l dominating set if and only if D ∩ Bl (v) 6= ∅ for all v ∈ V (H). Note that distance-1 dominating sets are the usual dominating sets. The minimum size of a distance-l dominating set in a hypergraph H is the distance-l domination number γd (H, l). Let further nd (k, γ, l) denote the minimum number of vertices that a k-uniform hypergraph H with no isolated vertices can contain if γd (H, l) ≥ γ holds. The next proposition shows that nd (k, γ, l) does not depend on l once l ≥ 2 is supposed. Proposition 1.2. For any k, l ≥ 2 and γ ≥ 1 we have nd (k, γ, l) = kγ, and the unique extremal hypergraph consists of γ pairwise disjoint edges. Proof. It is clear that the k-uniform hypergraph with just γ disjoint edges yields the upper bound nd (k, γ, l) ≤ kγ. We prove the lower bound by induction on γ. The case γ = 1 is trivial. So assume that γ ≥ 2, and let H = (V (H), E(H)) be a k-uniform hypergraph with γd (H, l) ≥ γ. Consider an arbitrary v ∈ V (H). Any vertex in N (Bl−1 (v)) is distance-l dominated by v, therefore the k-uniform hypergraph H0 induced by the edge set {H ∈ E(H) : H ∩ Bl−1 (v) = ∅} covers all vertices of H not distance-l dominated by v. The assumption γd (H, l) ≥ γ implies γd (H0 , l) ≥ γ − 1 and thus using that |Bl−1 (v)| ≥ k for l ≥ 2 and by induction we obtain |V (H)| = |Bl−1 (v)| + |V (H0 )| ≥ k + (γ − 1)k = γk. 3
Strict inequality holds whenever v has degree at least two. The problem becomes more interesting when disconnected hypergraphs get excluded. Hence, for k ≥ 2 and l, γ ≥ 1 let ndc (k, γ, l) denote the minimum number of vertices that a k-uniform connected hypergraph H must contain if it has γd (H, l) ≥ γ. To state our main result concerning ndc (k, γ, l) we need to define the following function: ( l kγ + max{k, γ} if l is even, f (k, γ, l) := 2l+1 kγ if l is odd. 2 Theorem 1.3. (a) For any k, l ≥ 2 we have (2l + 1)k (2l + 1)(k + 1) ≤ ndc (k, 2, l) ≤ min , (l + 1)k . 2 2 (b) For any k ≥ 2, l ≥ 4 and γ ≥ 3 we have l−1 − 1 γ < ndc (k, γ, l) ≤ f (k, γ, l). k 2 (c) For any k ≥ 2 and γ ≥ 3 we have kγ ≤ ndc (k, γ, 2) ≤ kγ + max{k, γ}. (d) For any k ≥ 2 and γ ≥ 3 we have kγ ≤ ndc (k, γ, 3) ≤ 2kγ.
The remainder of the paper is organized as follows: we prove Theorem 1.1 in Section 2, and Theorem 1.3 in Section 3. Section 4 contains some final remarks, also including a general upper bound on γdc (H, l) as a function of l, the number of vertices, and the edge size.
2
Proof of Theorem 1.1
In this section we prove our bounds on n× (k, γ, s) and n(k, γ, s). First we verify the bound k + k 1−1/(γ−s+1) ≤ n× (k, γ, s). Observe that it is enough to prove the statement for s = 1, since for any hypergraph H we have γ× (H, s) − (s − 1) ≥ γ× (H, 1) as for any s-tuple
4
dominating set D of H and a s0 -subset D0 of D the set D \ D0 (s − s0 )-tuple dominates H. Consequently n× (k, γ, s) ≥ n× (k, γ − (s − 1), 1), which implies the statement. To see n(k, γ, 1) ≥ k + k 1−1/γ let H be a k-uniform hypergraph with γ(H) ≥ γ ≥ 2. Let G = (V (H), E) be the graph with (u, v) ∈ E if and only if no H ∈ E(H) contains both u and v. The γ ≥ 2 condition means that for any vertex v ∈ V (H) there exists a u such that no edge H ∈ E(H) contains both u and v, thus G does not contain any isolated vertices. Let us write n = |V (H)| = |V (G)| = k + x and let t be the number of edges in a largest matching M = (V (M ), E(M )) of G. Note that two distinct vertices u0 , v 0 outside V (M ) cannot be adjacent to two distinct endpoints u, v of an edge e ∈ E(M ) as the matching (M \ {e}) ∪ {(u, u0 ), (v, v 0 )} would contradict the maximality of M . Then either just one of u and v has neighbors outside M , or none of them have any, or they share their unique neighbor outside M . We denote by e(v) the (or an) endpoint of e whose ‘outside’ neighborhood in this sense contains the ‘outside’ neighborhood of the other endpoint, and let de(v) denote the size of Ne(v) \ V (M ). By the definition of γ = γ× (H, 1) = γ(H, 1) and G we have that for any set Γ of γ − 1 vertices in V (G) there is a vertex v ∈ V (G) which is connected by edges in E(G) to all the vertices of Γ. If Γ is a subset of V (G) \ V (M ), then the vertex which is adjacent to all vertices of Γ must be in V (M ), since M is maximal. By this we obtain X de(v) |V (G) \ V (M )| ≥ . γ−1 γ−1 e∈E(M )
Writing d := maxe∈E(M ) de(v) the above inequality yields tdγ−1 ≥ (k + x − 2t)γ−1 , and rearranging gives d≥
k + x − 2t
. 1 t γ−1 Let e ∈ E(M ) be an edge with de(v) = d, and let H be any hyperedge H ∈ E(H) containing e(v). Just as any hyperedge, H must avoid an endpoint of each edge in M , and H is disjoint from Ne(v) \ V (M ). Therefore, we obtain k + x = n ≥ d + t + k and thus x ≥ d + t. Plugging the previous inequality into this and rearranging yields: γ−2
1
t γ−1 (x − t + 2t γ−1 ) ≥ k + x. γ−2
Now using that x ≥ t γand t ≥ t γ−1 , we obtain that the left-hand side of the previous inequality is at most x γ−1 + x and therefore we have γ
x γ−1 + x ≥ k + x, 5
which proves the required lower bound. To prove the bound n(k, γ, s) ≤ k + (4 + o(1))k 1−1/(γ−s+1) we need a construction. This involves projective geometries or linear vector spaces over finite fields. We will use the n Gaussian or q-binomial coefficient k q that denotes the number of k-dimensional subspaces of a vector space of dimension n over Fq , i.e. Qk n−i+1 − 1) n i=1 (q := Q k i k q i=1 (q − 1) and we will omit q from the subscript when it is clear from the context. Let q be a prime power, t be any positive integer and U be a γ-dimensional vector space over Fq . Let E1 , E2 , . . . , Em be the 1-dimensional of U and U1 , U2 , . . . , Um the γ-dimensional γsubspaces subspaces of U , where m = γ1 q = γ−1 = q γ−1 + q γ−2 + · · · + 1. Let A1 , A2 , . . . , Am , B be q pairwise disjoint sets with B = {b1 , b2 , . . . , bm } and |Ai | = t for all 1 ≤ i ≤ m. Let us define Hq,γ,t = {H1 , H2 , . . . , Hm } by [ Aj . Hi := {bi } ∪ j:Ei 6≤Uj
We claim that γ(Hq,γ−s+1,t , s) ≥ γ. Suppose not and let D = DB ∪ DA be a minimal sdominating set of H = Hq,γ−s+1,t with DB = D ∩ B, DA = D \ DB and |D| < γ. As every vertex d ∈ DB is contained in exactly one hyperedge Hd of H, each such d can be replaced by a vertex d0 ∈ V (H) \ (D ∪ B) to obtain an s-dominating set D0 with D0 ⊆ V (H) \ B 0 0 00 and |D T | = |D| < γ. Let D = {d1 , d2 , . . . , dp } and D = {d1 , d2 , . . . , dγ−s }. Then for Z = j:∃v∈D00 ∩Aj Uj we obtain dim(Z) ≥ 1. If E is a 1-subspace of Z, then the corresponding vertex b ∈ V (H) is not dominated by any vertex d ∈ D00 and thus at most (s − 1)-dominated by D0 , which is a contradiction. Let us consider the other parameters of the above hypergraph: n = |V (Hq,γ−s+1,t )| = m(t + 1) and Hq,γ−s+1,t is kq,γ−s+1,t -uniform with kq,γ−s+1,t = 1 + q γ−s t, therefore if t = q, then we obtain n = q γ−s+1 + 2(q γ−s + q γ−s−1 + · · · + q) + 1 and kγ = kq,γ−s+1,q = 1 + q γ−s+1 , 1−1/(γ−s+1) thus we have n ≤ kγ−s+1 + 4kγ−s+1 . This finishes the proof of the upper bound if k is one larger than the (γ − s + 1)st power of a prime. Finally, let us consider the general case when k 0 = 1 + q γ−s+1 + e with e < q 0γ−s+1 − q γ−s+1 where q 0 is the smallest prime larger than q. It is well-known that q 0 = q + o(q) and e e, all thus e = o(q γ−s+1 ). Let C1 , C2 , . . . , Cq+1 be pairwise disjoint sets all of size d q−γ+s+2 being disjoint from V (Hq,γ−s+1,q ). We renumber the subspaces U1 , U2 , . . . , Um in such a way that U1 , U2 , . . . , Uq+1 correspond to the dual of a (q + 1)-arc in P G(γ − s, q), i.e. every 1subspace E of V is contained in at most γ − s − 1 subspaces among U1 , U2 , . . . , Uq+1 . (For a general introduction to finite geometries, see [15].) Therefore, for any 1 ≤ i ≤ m, the sets 6
Ii := S {j : Ei 6≤ Uj , 1 ≤ j ≤ q + 1} satisfy |Ii | ≥ q − γ + s + 2 and thus there exists a set Ti ⊂ j∈Ii Cj of size e. Let us define [
Hi0 := {bi } ∪
Aj ∪ Ti .
j:Ei 6≤Uj
By definition we have |Hi0 | = k 0 for all i = 1, 2, . . . , m. The s-domination number of the new hypergraph is the same as that of the old one, as for any v ∈ Ci and u ∈ Ai we have Nu ⊂ Nv . Moreover the number n0 of vertices in the new hypergraph is n+d
e e(q + 1) ≤ k + 4k 1−1/(γ−s+1) + e + Oγ (e/q) ≤ k 0 + (4 + o(1))k 01−1/(γ−s+1) , q−γ+s+1
as Oγ (e/q) = o(q γ−s ) holds by e = o(q γ−s+1 ).
3
Distance domination
In this section we prove Theorem 1.3, the lower and upper bounds on ndc (k, γ, l).
3.1
The j-radius of trees
We start with some definitions and an auxiliary statement that we will use in the proof. Definition. For positive integers a1 , a2 , ..., ah the spider graph, denoted by S(a1 , a2 , . . . , ah ), P is the tree on 1 + hi=1 ai vertices which is obtained from h paths of lengths a1 , a2 , . . . , ah , respectively, by identifying the first vertices of those paths to a single vertex v of degree h. Hence, S(a1 , a2 , . . . , ah ) \ {v} has h connected components, say C1 , C2 , . . . , Ch , where each Ci is a path Pai on ai vertices (for i = 1, 2, ..., h). In a connected graph G = (V (G), E(G)), the excentricity of a vertex v ∈ V (G) is defined as excG (v) := max{dG (u, v) : u ∈ V (G)} and let the radius of G is r(G) := min{excG (v) : v ∈ V (G)}. More generally, for any ∅ = 6 W ⊂ V (G) let us write excG (W ) := max{min{dG (u, w) : w ∈ W } : u ∈ V (G)} 7
and for an integer j ≥ 1 let the j-radius of G be rj (G) := min{excG (W ) : W ⊂ V (G), |W | ≤ j}. Certainly we have r(G) = r1 (G). Finally, let rj (n) := max{rj (T ) : |V (T )| = n, T is a tree}. The numerical bounds themselves in the next lemma concerning the radius of a tree are folklore; for later use, however, we need a more detailed assertion which describes some structural properties, too. Some bounds on the function rj (n) can be derived also from results of Meir and Moon [17], but the following is a little sharper. Lemma 3.1. Let n ≥ j be positive integers. Then we have n n ≤ rj (n) ≤ . j+1 j+1 Moreover, r1 (n) = n−1 and 2 e is the path Pn on n vertices. (i) if n is even, then the only tree with r1 (T ) = d n−1 2 (ii) If n is odd and r1 (T ) = d n−1 e holds, then T is a path Pn−1 with a pendant edge. 2 Furthermore, T contains two copies of Pn−1 if and only if T is either a path Pn or a fork Fn . Otherwise T contains just one copy of Pn−1 . Proof. Let us first prove the statements about r1 (n). Let T be an arbitrary tree on n vertices and let v be a middle vertex of a longest path P in T . If P contains l vertices, then any e from v. This implies all assertions of the lemma if n is vertex is at distance at most d l−1 2 even. If n is odd, this implies that T must contain a path on n − 1 vertices and thus T is a path Pn−1 and a pendant edge. Let us now prove the general lower bound. We claim that n−1 n n+j−1 n = rj S , ,..., j+1 j+1 j+1 j+1 holds, which proves the lower bound by the definition of rj (n). To see that the claim is true, n observe that any set U ⊂ V (S(b n−1 c, b j+1 c, . . . , b n+j−1 c)) of size j is disjoint from at least j+1 j+1 n+j−1 n−1 n one component C of S(b j+1 c, b j+1 c, . . . , b γ+1 c) \ {v}. n c, b j+1 c, . . . , b n+j−1 c) belonging to C has distance Thus if v ∈ / U , then the leaf of S(b n−1 j+1 j+1 at least n−1 n 1+ ≥ j+1 j+1 8
from any vertex of U . If v ∈ U holds, then U is disjoint from at least two components C1 , C2 of n n S(b n−1 c, b j+1 c, . . . , b n+j−1 c) \ {v}, and the leaf of S(b n−1 c, b j+1 c, . . . , b n+j−1 c) belonging to j+1 j+1 j+1 j+1 n the larger path has distance at least b j+1 c from v and thus from U . This completes the proof of the general lower bound. To see the general upper bound, let T be any tree on n vertices. We will use the following claim repeatedly. Claim 3.2. Let m < n be two positive integers. Then in any tree T on n vertices there exists a vertex v such that if C1 , C2 , . . . , Cs denote Ps those components of T \ {v} whose all vertices are at distance at most m from v, then i=1 |Ci | ≥ m holds. Proof of Claim 3.2. Let P be a longest path of T . If P contains at most m vertices, then any vertex can play the role of v. If P contains at least m + 1 vertices, then let v be the (m + 1)st vertex from one end of P . For t = 1, 2, . . . , j − 1 let mt = b n+t−1 c and let T1 = T . We apply Claim 3.2 to Tt and j+1 mt for t = 1, 2, . . . , j − 1 to obtain vt ; and then set t Tt+1 := Tj \ ∪ki=1 Ci,t ,
where the Ci,t (i = 1, 2, ..., kt ) are the components of Tt \ {vt } whose vertices are at distance Pt at most mt from vt . By the claim we also have ki=1 |Ci,t | ≥ mt . n In this way we obtain a tree Tj of at most d2 j+1 e vertices. Let vj be a vertex of Tj |V (T )|−1
j e from all vertices of Tj . Such a vertex exists by the result on within distance d 2 n r1 (n). Clearly, U = {v1 , v2 , . . . , vj } is a set of vertices with excT (U ) ≤ d j+1 e, which proves n rj (T ) ≤ d j+1 e.
3.2
Putting things together: the proof of Theorem 1.3
Let us first prove the upper bounds of Theorem 1.3. To do so we introduce two types of hypergraphs with distance-l domination number γ. The second construction will prove the upper bounds of (b), (c), and (d). If γ = 2, then the construction giving the smaller number of vertices depends on the values of k and l. This is why we have the minimum of two expressions in the upper bound of (a). Construction 1: For i = 1, . . . , 2l(γ − 1) + 1 let Ui be pairwise disjoint sets, and let vi and w be distinct S vertices which are not elements of 2l(γ−1)+1 Ui . During Construction 1 all the indices will i=1 be taken modulo 2l(γ − 1) + 1, e.g. we then have 2l(γ − 1) + 2 = 1.
9
If k is odd, let |Ui | = following way. Let
k−1 2
for all i. We define a hypergraph H = (V (H), E(H)) in the 2l(γ−1)+1
V (H) :=
[
(Ui ∪ {vi }),
i=1
and let the hyperedges of H be Hi := Ui ∪ Ui+1 ∪ {vi } for i = 1, . . . , 2l(γ − 1) + 1. Then the size of V (H) is (2l(γ − 1) + 1)(k + 1) . 2 If k is even, let |U2i | = k2 for i = 1, . . . , l(γ − 1) and |U2i+1 | = We define H with the vertex set
k 2
− 1 for i = 0, . . . , l(γ − 1).
2l(γ−1)+1
V (H) := {w} ∪
[
(Ui ∪ {vi }),
i=1
and with the edge set E(H) := {Hi | 1 ≤ i ≤ 2l(γ − 1) + 1}, where Hi := Ui ∪ Ui+1 ∪ {vi } if i = 1, . . . , 2l(γ − 1), and Hi := Ui ∪ Ui+1 ∪ {vi , w} if i = 2l(γ − 1) + 1. Then, (2l(γ − 1) + 1)(k + 1) 1 (2l(γ − 1) + 1)(k + 1) |V (H)| = + = . 2 2 2 To see that γd (H, l) ≥ γ holds in both cases, observe the following facts: • vertex vi distance-l dominates a vertex vj exactly for j ∈ {i − l + 1, ..., i + l − 1}, • vertex w distance-l dominates a vj exactly for j ∈ {2l(γ − 1) − l + 2, ..., 2l(γ − 1) + l}, • a vertex u ∈ Ui distance-l dominates a vj exactly for j ∈ {i − l, ..., i + l − 1}. 10
So, every vertex in V (H) distance-l dominates at most 2l vertices vi . This yields γd (H, l) ≥ γ. Construction 2: This construction relies on the spider graph S = S(a1 , a2 , . . . , aγ ) with all of the ai being equal to bl/2c. Let v be the only vertex of S with degree γ. Let u1 , u2 , . . . , uγ be the neighbors of v in S, and let u01 , u02 , . . . , u0γ be the vertices of S that are at distamce bl/2c from v. Let W be a set of size max{k, γ}. Take a partition (W1 , W2 , . . . , Wγ ) of W such that |Wi | = b |W |+i−1 c. Finally, for every u ∈ V (S) \ {v}, let Uu := Uu,1 ∪˙ Uu,2 be a set of size k γ such that • u ∈ Uu,1 holds for all u ∈ V (S) \ {v}, • Uu ∩ Uu0 = ∅ holds for all u 6= u0 ∈ V (S) \ {v}, • Uu ∩ W = ∅ holds for all u ∈ V (S) \ {v}, • |Uu,1 | = |Wi | for all those u ∈ V (S) \ {v} which lie in the same component of S \ {v} as ui . With the help of the previously defined sets we construct a k-uniform hypergraph H in the following way, depending on the parity of l: Case I: l is even Let the vertex set of H be V (H) = W ∪ |V (H)| =
S
u∈V (S)\{v}
Uu . Thus we have
klγ + max{k, γ}. 2
The edge set E(H) contains the following four types of hyperedges: 1. all k-subsets of W , i.e. Wk ⊂ E(H), 2. for all u ∈ V (S) \ {v}, we have Uu ∈ E(H), 3. for all i = 1, 2, . . . , γ let Wi ∪ Uui ,2 ∈ E(H), 4. for every edge (u, u0 ) = e ∈ E(S) with u, u0 6= v if dS (u, v) < dS (u0 , v) holds, then let Uu,1 ∪ Uu0 ,2 ∈ E(H). Clearly, H is connected due to Wk ⊂ E(H). We claim that γd (H, l) ≥ γ holds. Indeed, if D ⊂ V (H) has size at most γ − 1, then there exists an i ≤ γ such that [ D ∩ (Wi ∪ Uu ) = ∅ u∈Ci
11
Figure 1: Construction 2 in case of even l
holds where Ci is the component of S \ {v} containing ui . Then u0i is at distance at least 1 + 2 2l = l + 1 from any vertex of D and thus u0i is not distance l-dominated by D. Case II: l is odd In addition to the sets defined above, let Z1 , Z2 , . . . , Zγ be pairwise disjont sets of size k − |Wi |, each of which is disjoint from all previously defined sets. Let the vertex set of H be γ [ [ V (H) = W ∪ Uu ∪ Zi . u∈V (S)\{v}
i=1
Thus we have
l |V (H)| ≤ d ekγ. 2 As for the edge set of H, there is a fifth type of hyperedge: 5. for all 1 ≤ i ≤ γ let Uu0i ,1 ∪ Zi ∈ E(H). The fact that γd (H, l) ≥ γ follows similarly as in the previous case, because for any (γ −1)-set D ⊂ V (H) there exists an i such that any vertex z ∈ Zi is at distance at least l + 1 from D. Let us now turn our attention to the lower bounds. We prove first that of (a). Consider a connected k-uniform hypergraph H with γd (H, l) ≥ 2. Let M be a maximal matching in 12
H obtained in the following way. Let M1 := {H1 }, I1 := {H ∈ E(H) \ {H1 } : H ∩ H1 6= ∅} and R0 := E(H) \ (M1 ∪ I1 ). Then for s ≥ 2 we define a sequence Ms , Is , Rs of partitions of E(H) such that: 1. Ms is a matching, 2. every hyperedge in Is meets at least one hyperedge in Ms , and 3. all hyperedges in Rs are disjoint from all hyperedges in Ms . If Ms , Is , Rs are defined with Rs 6= ∅, then let Hs+1 ∈ Rs be a hyperedge such that Hs+1 ∩ Is 6= ∅ for some Is ∈ Is . The existence of such Hs+1 follows from the assumption that H is connected. Set Ms+1 := Ms ∪ {Hs+1 }, Is+1 := Is ∪ {R ∈ Rs \ {Hs+1 } : R ∩ Hs+1 6= ∅} and Rs+1 := E(H) \ (Ms+1 ∪ Is+1 ). For the smallest positive t with Rt = ∅, we let M := Mt . Thus the size of M is t. Now let us consider the auxiliary graph GM with vertex set M and e = {Hi , Hj } ∈ E(GM ) if and only if there exists H ∈ H with H ∩ HS i 6= ∅ and H ∩ Hj 6= ∅. By the definition of M, the graph GM is connected. For a vertex v ∈ H∈M H let Hv denote the only element of M containing v. Suppose that for a pair H, H 0 ∈ M we have dGM (H, H 0 ) = r. Then for any pair of vertices u ∈ H, v ∈ H 0 we have dH (u, v) ≤ 1 + 2r. To see this, consider the sequence H, He1 , Hi1 , He2 , Hi2 , . . . , Her , H 0 , where es is the sth edge in a shortest path from H to H 0 and His is the sth vertex (i.e. a hyperedge in H) in the same path. By the maximality of M, for every vertex w of H there exists an edge Hw containing w and an edge H ∈ M with Hw ∩ H 6= ∅, therefore by the observation above we have dH (u, w) ≤ 2 + 2rGM (u) for every u ∈
S
H∈M
H and w ∈ V (H).
If t ≥ l + 1 holds, then |V (H)| ≥ kt ≥ k(l + 1), proving the desired lower bound. Now suppose that t ≤ l − 2 or t = l − 1 with t being odd. As we have noted, GM is connected and thus by Lemma 3.1 we obtain t−1 . r(GM ) ≤ 2 e holds and so, by Therefore, there exists an H ∗ ∈ M = V (GM ) such that rGM (H ∗ ) ≤ d t−1 2 ∗ the above, for a vertex v ∈ H we have t−1 0 dH (v, v ) ≤ 2 + 2 2 13
for any vertex v 0 ∈ V (H). So in this case a vertex v ∈ H ∗ distance-l dominates H, contradicting γd (H, l) ≥ 2. If t = l − 1 and t is even, then let T be a spanning tree of GM . By Lemma 3.1 we obtain that T is a path on t vertices. So we may assume that E(GM ) ⊃ {(Hi , Hi+1 ) : i = 1, . . . , t − 1}. Let e = (Ht/2 , Ht/2+1 ) and consider a vertex v ∈ Ht/2 ∩He . As for vertices v 0 with Hv0 ∩Hi 6= ∅ for some i > t/2, a shortest path in H between v and v 0 need not contain Ht/2 . Thus we obtain that v distance-l dominates H, contradicting γd (H, l) ≥ 2. Finally, it remains S to prove the lower bound of (a) in case of t = l and thus it is enough to prove that |V (H) \ H∈M H| ≥ k/2 holds. We may and will assume that the radius of GM is d l−1 e. Let T be a spanning tree of GM . By Lemma 3.1 we know that T is a path if l is even, 2 and T contains a path on l − 1 vertices if l is odd. We claim that even if l is odd, T must be a path on t vertices. Indeed, otherwise any vertex v ∈ He distance-l dominates H where e is the middle edge of a path on l − 1 vertices that is contained in T . This would contradict γd (H, l) ≥ 2. By this we may assume that E(GM ) ⊃ {(Hi , Hi+1 ) : i = 1, . . . , l − 1}. Claim 3.3. We have the following: (i) For any pair of edges e, e0 in T we have He ∩ He0 = ∅. S (ii) There exist w, w0 ∈ V (H) \ H∈M H and Hw , Hw0 ∈ E(H) with w ∈ Hw and w0 ∈ Hw0 , such that Hw meets only H1 and Hw0 meets only Hl , moreover Hw and Hw0 are disjoint from all the other H ∈ M and also from He for all e ∈ E(T ). Proof of Claim. We have two cases depending on the parity of l. Case I: l is even. Now we prove (i) in this case. Suppose that Hei ∩ Hej 6= ∅ with ei = (Hi , Hi+1 ), ej = (Hj , Hj+1 ). If i < j ≤ l/2, then a vertex v ∈ Hel/2 ∩ Hl/2+1 distance-l dominates H, contradicting γdist (H, l) ≥ 2.. Similarly, if i < j and j ≥ l/2, then a vertex v ∈ Hel/2 ∩ Hl/2 distance-l dominates H, contradicting γdist (H, l) ≥ 2. Also, if i < l/2 < j, then if l/2 − i ≤ j −l/2, then a vertex v from Hl/2−1 ∩Hel/2−1 distance-l dominates H, while if l/2−i ≥ j −l/2, then a vertex v from Hl/2+2 ∩ Hel/2+1 distance-l dominates H, contradicting γdist (H, l) ≥ 2. We are done with (i) in Case I. S To see (ii) suppose that, for every w ∈ V (H) \ H∈M H and Hw containing w, the hyperedge Hw meets He for some e ∈ E(T ) or Hw meets some Hz with z ≥ 2. Then a vertex in Hen/2 ∩ Hn/2+1 distance-l dominates H, contradicting γdist (H, l) ≥ 2. The existence of w0 and Hw0 can be shown analogously. This proves (ii) in Case I. 14
Case II: l is odd. The proof of this case is S very similar to the previous one. Let us just show (ii). Suppose that, for every w ∈ V (H) \ H∈M H and Hw containing w, the hyperedge Hw meets He for some e ∈ E(T ) or Hw meets some Hz with z ≥ 2. Then a vertex in Hedn/2e ∩ Hdn/2e distance-l dominates H, contradicting γdist (H, l) ≥ 2. S S Note that Hw ∩ Hw0 ⊂ V (H) \ H∈M H and also Hw ∪ Hw0 ∪ e∈E(T ) He ⊂ V (H), and thus writing I = |Hw ∩ Hw0 | we obtain |V (H)| ≥ max{lk + I, (l + 1)k − I} ≥ lk + k/2. This finishes the proof of the lower bound of (a). Next we prove the lower bound of (b). We will need the following lemma. Lemma 3.4. For any γ, l ≥ 2, let t∗ denote smallest t with rγ−1 (t) ≥
l−1 . 2
Then we have
ndc (k, γ, l) ≥ t∗ k. Proof. Let H be a connected k-uniform hypergraph with γd (H, l) ≥ γ. Let M be a maximal matching in H obtained as in the proof of the lower bound of part (a), and let us consider the auxiliary graph GM . For a vertex v ∈ ∪H∈M H let Hv denote the only element of M . containing v. Let the size of M be t. We assume first that t < t∗ , what means rγ−1 (t) < l−1 2 Suppose that for a pair H, H 0 ∈ M we have dGM (H, H 0 ) = r. Then for any pair of vertices u ∈ H, v ∈ H 0 we have dH (u, v) ≤ 1 + 2r. To see this, consider the sequence H, He1 , Hi1 , He2 , Hi2 , . . . , Her , H 0 , where es is the sth edge in a shortest path from H to H 0 and His is the sth vertex (i.e. a hyperedge in H) in the same path. Let U ⊂ M be a subset of size γ − 1 with rGM (U) = rγ−1 (GM ) ≤ rγ−1 (t), and let L ⊂ V (H) be a set containing one vertex from each U ∈ U. By the maximality of M, for every vertex w of H there exists an edge Hw containing w and an edge H ∈ M with Hw ∩ H 6= ∅. Therefore by the observation above and by the definition of U, there exists a U ∈ U and a vertex u ∈ U for which we have dH (u, w) ≤ 2 + 2rGM (U ) ≤ 2 + 2rγ−1 (t) < 2 + 2
l−1 = l + 1. 2
This means that if t < t∗ holds, then the (γ − 1)-subset L distance-l dominates H. Therefore M consists of at least t∗ hyperedges and thus |V (H)| ≥ t∗ k holds. The lower bound of (b) follows by applying Lemma 3.1 with j = γ − 1 together with Lemma 3.4, noting that d γt e ≥ l−1 implies γt > l−1 − 1. 2 2 Finally, we prove the lower bound of (c) and (d). This will follow from the claim that any maximal matching in the edge set E(H) of a connected hypergraph H with γd (H, 2) ≥ γ has size at least γ. To see this suppose that M = {H1 , H2 , . . . , Hm } is a maximal matching 15
in E(H) and for any i = 1, 2, . . . , m let vi be a vertex of Hi . As any vertex v ∈ V (H) is contained in a hyperedge Hv which, by maximality of M, intersects some Hi ∈ M, the set D = {vi : i = 1, 2, . . . , m} distance-2 dominates H. Therefore m ≥ γ must hold as claimed.
4
Final remarks and open problems
We addressed the problem of finding the minimum number of vertices that a connected kuniform hypergraph with high domination number must contain, and we considered two main variants of the problem. For the original notion of domination and for s-wise domination we found general lower and upper bounds on n(k, γ, s) in which even the order of magnitude of the second term matches. The natural open problem occurs: it can be of interest to find the constant coefficient of this second term. Theorem 1.3, our main result concerning distance domination determines the asymptotics of ndc (k, γ, l) if k and γ are fixed and l tends to infinity, or if all three parameters tend to infinity. Closing the gap of roughly 2kγ between the upper and lower bounds remains an interesting open problem. We had a good reason to choose the notion of Berge paths in the definition of distance-l domination. The most common other definitions of a path in hypergraphs are linear paths, where two consecutive hyperedges of the path must share exactly one vertex (an even more restrictive notion is a loose path) and tight paths where the vertices v1 , v2 , . . . , vk+l−1 of the path should be chosen in such a way that the ith hyperedge of the path is {vi , vi+1 , . . . , vi+k−1 } for all i = 1, 2, . . . , l. This implies that consecutive hyperedges of a tight path share k − 1 vertices. Note that in the construction showing the upper bound of Theorem 1.1 no pair of hyperedges has intersection size 1 or k − 1, therefore the construction does not contain linear or tight paths of length larger than 1 and thus distance domination would not differ from ordinary domination, had we used these notions of hypergraph paths to define distance. There are various results on different domination numbers of a hypergraph in the literature: on the s-domination number in [2], on the inverse domination number in [16], on the total domination number in [5], and on the connection of the domination number with the transversal number in [3], [4]. Let us finish with the following theorem that can be obtained simply by rearranging the lower bound of Theorem 1.3. In the style of Meir and Moon [17], it uses only the size of the vertex set, the prescribed distance bound l, and the uniformity of H. Theorem 4.1. If H is a connected k-uniform hypegraph with |V (H)| = n, then ( n if l = 2, 3 or 4, γdc (H, l) ≤ nk 2 · if l > 4. k l−3 It remains an open problem to make these upper bounds tight. 16
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